Briefly, floating point numbers are a way to represent a wide numeric range in a limited set of bits by storing a limited number of significant digits. As the absolute size of the number increases, the absolute precision decreases.

The principle is similar to exponential notation for numbers (e.g. 3.523 x 106) except that IEEE floating point values use powers of 2 instead of 10.

Binary Format

The standard defines two precisions - single is a 32-bit representation and double is a 64-bit representation. These typically correspond to the C types float and double on platforms where the underlying hardware uses IEEE 754 (which is to say most current hardware architectures).

Each value is split into three chunks of bits, with byte boundaries ignored:

Section

Single

Double

Meaning

Sign

1 bit

1 bit

Set if the number is negative

Exponent

8 bits

11 bits

The power of 2 to multiply

Significand

23 bits

52 bits

The sigificant digits of the number

Note that the significand is also known as the mantissa in some texts, although this is discouraged by the IEEE 754 standards committee and others because of confusion with other uses of the term.

Sign bit

This single bit is 0 for positive numbers and 1 for negative numbers. Note that the sign bit is valid in most cases even for special values - for example, the standard differentiates between positive and negative zero.

Exponent

The exponent indicates the power of 2 by which the significand is multiplied. It is stored in biased form, which is an easy way to store a signed value in an unsigned field by simply adding a fixed value. The range of the significant, and the bias value which is added to it to obtain the actual unsigned value stored, is:

Precision

Bias

Range of valid exponents

Single

127

-126 – 127

Double

1023

-1022 – 1023

Note that the range is missing the values at each end — this is because a zero exponent and an exponent with all bits set both have special meanings.

Significand

This portion of the value stores the significant binary digits. To save space, there is assumed to be a leading 1 digit. For example, the significand 1.0100111… is stored as 0100111…. This form is said to be normalised.

Note that this is a simplification as there is also a denormalised form for values near zero, which is described below.

Normalised Values

The most common form of IEEE floating point numbers is the normalised form — this is where the exponent has a value in the valid range (once the bias has been subtracted from the unsigned value stored). As explained above, the significand stores only the digits after the leading 1, which is implicit.

If the standard C library is available, the frexp() function normalises a floating point value such that the fractional part will be in the range 0.5 ≤ x < 1.0. Multiplying this value by 2 and reducing the exponent by 1 yields a value in the desired range 1.0 ≤ x < 2.0. At this point the leading digit can then be discarded as the implicit leading 1 (see the Significand section for details).

If frexp() is available then it should be used, as it will likely use the underlying hardware representation to avoid expensive loops. However, a naive implementation can quite simply mimic its functionality — the following version demonstrates the principle, but a production version would also need to check for special values (zero, NaN, infinities) as well as catching under- and overflows:

Denormalised Values

At the lower end of the scale, very small numbers can be stored in denormalised form, where the implicit leading digit is a 0 instead of 1. In IEEE 754 this is represented by an exponent field of all zeroes and a non-zero significand. The actual exponent that this value represents is one higher than would be expected from a zero exponent field:

Precision

Denormalised Exponent

Single

-126

Double

-1022

At first sight it appears that this introduces overlap with the normalised numbers, as these are the lowest value exponents for a normalised value. However, the leading zero in the significand means that in fact there's no overlap.

Special Values

Aside from normalised and denormalised numbers, there are a variety of values represented by specific bit patterns in the representation.

Zero

Sign bit

Any

Exponent

Zero

Significand

Zero

A value of exactly zero is represented by a exponent and significand of zero. The sign bit may be set or unset and IEEE 754 has the concept of both a positive and negative zero. For standard comparisons, however, these will both compare equal with zero, so the comparison -0.0 < 0.0 yields false.

To determine the sign of a floating point value including zero, the copysign() function can be used with a non-zero value, or the signbit() macro can be used more directly on some platforms (not available on WinCE, for example).

Infinity

Sign bit

Any

Exponent

All bits set

Significand

Zero

If all bits are set in the exponent and the signficand is zero then the value represented is either positive or negative infinity, depending on the sign bit.

NaN

Sign bit

Any

Exponent

All bits set

Significand

Non-zero

If all bits are set in the exponent and the significand is non-zero then the value represented is not a number often abbreviated to NaN. This is a special range of values which are returned by operations that don't yield a valid arithmetic value.

Since any non-zero significand value is permitted, this allows a range of values to be specified. The sign bit may also be set or unset, which could be used to differentiate between different types of NaN. The main distinction is between a quiet NaN which has the most-significant bit of the significand set and a signalling NaN which has the MSB of the significand clear1) (although the overall value must still be non-zero).

The intention of the signalling NaN is that this will raise some sort of exception, and then go on to yield a quiet NaN if a result is required. This means that each error is only raised once. However, the support for this on various platforms seems to differ.